论文标题
曲线复合物中有效的大地测量及其点图
Efficient geodesics in the curve complex and their dot graphs
论文作者
论文摘要
对于属于$ g $的封闭式表面的曲线的复合物，$ \ mathcal {c}（s_ {g> 1}）$，在Arxiv：1408.4133中引入了有效的测量概念。在那里，已经确定在任何两个顶点（$v_α，v_β\ in \ Mathcal {c}（c}（s_g）$）之间，总是存在（有限的许多）有效的大地测量学，代表简单的封闭曲线的同质类别，$α，β\ subset s_g $。用于建立有效地测量的存在的主要工具是一个点图，它是一个预订方案，用于记录参考弧的相交模式，$γ\ subset s_g $，其简单的封闭曲线与零骨架中的地理路径的顶点相关，$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mathcal {c}^0（s_g gg）特别是，对于长度$ d \ geq 3 $的$v_α$和$v_β$之间的有效测量，这表明，与$v_α$相对于顶点的任何曲线，最多可在$v_α$中距离$v_α$最多$ d -2 $ d -2 $倍。在本说明中，我们对在有效的测量边缘路径中整个顶点的点图的表征“形状”进行了更广泛的研究。这项研究的关键是，对于任何有效的大地测量，都包含在主轴形状区域内的点图。由于$ s_g $的任何曲线的尼尔森 - 瑟斯顿坐标直接从其与有限的许多参考弧相交数中得出，因此纺锤形点图控制与有效地理位置的曲线相关的曲线的坐标行为。
For the complex of curves of a closed orientable surface of genus $g$, $\mathcal{C}(S_{g>1})$, the notion of efficient geodesic in was introduced in arXiv:1408.4133. There it was established that there always exists (finitely many) efficient geodesics between any two vertices, $ v_α , v_β \in \mathcal{C}(S_g)$, representing homotopy classes of simple closed curves, $α, β\subset S_g$. The main tool for used in establishing the existence of efficient geodesic was a dot graph, a booking scheme for recording the intersection pattern of a reference arc, $γ\subset S_g$, with the simple closed curves associated with the vertices of geodesic path in the zero skeleton, $\mathcal{C}^0(S_g)$. In particular, for an efficient geodesic between $v_α$ and $v_β$ of length $d \geq 3$, it was shown that any curve corresponding to the vertex that is distance one from $v_α$ intersects any $γ$ at most $d -2$ times. In this note we make a more expansive study of the characterizing "shape" of the dot graphs over the entire set of vertices in an efficient geodesic edge-path. The key take away of this study is that the shape of a dot graph for any efficient geodesic is contained within a spindle shape region. Since the Nielson-Thurston coordinates of any curve on $S_g$ are directly derived from its intersection number with finitely many reference arcs, spindle shaped dot graphs control the coordinate behavior of curves associated with the vertices of an efficient geodesic.
